While Bayesians do not like classical P-values and prefer measuring evidence in data through posterior probabilities of parameters or models, some problems like testing or exploration of goodness of fit of a single given model have led to the introduction of P-values. We confine ourselves to this particular context of goodness of fit in the brief discussion of Bayesian P-values. Most of this material is taken from Ghosh, Delampady and Samanta (2006) and Ghosh, Purkayastha and Samanta (2005).
Suppose that we have a single model M that specifies a density f(x | θ), θ ∈ Θ for the observable X and the Bayesian has a prior π(θ). The Bayesian wishes to examine how well the model M fits the data x obs on the basis of a statistic T(X) which measures the goodness of fit of data and model. Of course, T is also chosen by the Bayesian even though it is not part of the usual paradigm for Bayesian inference.
Let
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References and Further Reading
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Ghosh, J.K., Delampady, M. (2011). Bayesian P-Values. In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_136
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